theorem Th19:
  i in dom b1 implies b1/.i|-- b1 = Line(1.(K,len b1),i)
proof
  set ONE=1.(K,len b1);
  set bb=b1/.i|-- b1;
  consider KL be Linear_Combination of V1 such that
A1: b1/.i = Sum(KL) & Carrier KL c= rng b1 and
A2: for k st 1<=k & k<=len bb holds bb/.k=KL.(b1/.k) by MATRLIN:def 7;
  reconsider rb1=rng b1 as Basis of V1 by MATRLIN:def 2;
A3: rb1 is linearly-independent by VECTSP_7:def 3;
  b1/.i in {b1/.i} by TARSKI:def 1;
  then b1/.i in Lin{b1/.i} by VECTSP_7:8;
  then consider Lb be Linear_Combination of {b1/.i} such that
A4: b1/.i=Sum Lb by VECTSP_7:7;
  assume
A5: i in dom b1;
  then
A6: b1.i=b1/.i by PARTFUN1:def 6;
  then
A7: Carrier Lb c= {b1.i} by VECTSP_6:def 4;
A8: b1.i in rb1 by A5,FUNCT_1:def 3;
  then {b1.i}c= rb1 by ZFMISC_1:31;
  then Carrier Lb c= rb1 by A7;
  then
A9: Lb = KL by A4,A1,A3,MATRLIN:5;
A10: width ONE=len b1 by MATRIX_0:24;
A11: Indices ONE=[:Seg len b1,Seg len b1:] by MATRIX_0:24;
A12: len b1=len bb by MATRLIN:def 7;
A13: b1/.i<>0.V1 by A6,A3,A8,VECTSP_7:2;
A14: now
    let j such that
A15: 1<=j & j<=len bb;
A16: j in Seg len b1 by A12,A15;
    i in Seg len b1 by A5,FINSEQ_1:def 3;
    then
A17: [i,j] in Indices ONE by A11,A16,ZFMISC_1:87;
A18: j in dom b1 by A12,A15,FINSEQ_3:25;
A19: dom bb=dom b1 by A12,FINSEQ_3:29;
    now
      per cases;
      suppose
A20:    i=j;
        Lb.(b1/.i) *(b1/.i) = b1/.i by A4,VECTSP_6:17
          .= 1_K*(b1/.i);
        then
A21:    1_K = KL.(b1/.i) by A13,A9,VECTSP10:4
          .= bb/.j by A2,A15,A20;
        1_K = ONE*(i,j) by A17,A20,MATRIX_1:def 3
          .= Line(ONE,i).j by A10,A16,MATRIX_0:def 7;
        hence Line(ONE,i).j=bb.j by A18,A19,A21,PARTFUN1:def 6;
      end;
      suppose
A22:    i<>j;
        b1 is one-to-one by MATRLIN:def 2;
        then b1.i <> b1.j by A5,A18,A22;
        then
A23:    not b1.j in Carrier Lb by A7,TARSKI:def 1;
A24:    0.K = ONE*(i,j) by A17,A22,MATRIX_1:def 3
          .= Line(ONE,i).j by A10,A16,MATRIX_0:def 7;
        b1.j = b1/.j by A18,PARTFUN1:def 6;
        then 0.K = KL.(b1/.j) by A9,A23
          .= bb/.j by A2,A15;
        hence Line(ONE,i).j=bb.j by A18,A19,A24,PARTFUN1:def 6;
      end;
    end;
    hence Line(ONE,i).j=bb.j;
  end;
  len Line(ONE,i)=len b1 by A10,CARD_1:def 7;
  hence thesis by A12,A14;
end;
